Embedding cycles in finite planes

TL;DR

This paper investigates how cycles of various lengths can be embedded in finite affine and projective planes, establishing that all cycles within certain length ranges are embeddable in these geometrical structures.

Contribution

It proves that all cycles of length between 3 and q^2 (affine) or q^2+q+1 (projective) can be embedded in any finite affine or projective plane of order q.

Findings

All cycles of length 3 to q^2 embed in affine planes.

All cycles of length 3 to q^2+q+1 embed in projective planes.

The results hold for any finite plane of the specified order.

Abstract

We define and study embeddings of cycles in finite affine and projective planes. We show that for all $k$, $3\le k\le q^2$, a $k$-cycle can be embedded in any affine plane of order $q$. We also prove a similar result for finite projective planes: for all $k$, $3\le k\le q^2+q+1$, a $k$-cycle can be embedded in any projective plane of order $q$.